\c>*v 


C*mV  V. 


INTRODUCTION  TO  A GENERAL  THEORY 
OF  ELEMENTARY  PROPOSITIONS 


EMIL  L.  POST 


Submitted  in  Partial  Fulfilment  of  the  Requirements' for  the  Degree 
of  Doctor  of  Philosophy,  in  the  Faculty  of  Pure  Science, 
Columbia  University. 


Reprinted  from 

American  Journal  of  Mathematics 
Vol.  XLIII,  No.  3 


1921 


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INTRODUCTION  TO  A GENERAL  THEORY  OF  ELEMENTARY 

PROPOSITIONS. 

By  Emil  L.  Post. 

Introduction. 

In  the  general  theory  of  logic  built  up  by  Whitehead  and  Russell*  to 
furnish  a basis  for  all  mathematics  there  is  a certain  subtheory  f which  is 
junique  in  its  simplicity  and  precision;  and  though  all  other  portions  of  the 
work  have  their  roots  in  this  subtheory,  it  itself  is  completely  independent  of 
jthem.  Whereas  the  complete  theory  requires  for  the  enunciation  of  its 
propositions  real  and  apparent  variables,  which  represent  both  individuals 
and  propositional  functions  of  different  kinds,  and  as  a result  necessitates 
the  introduction  of  the  cumbersome  theory  of  types,  this  subtheory  uses 
only  real  variables,  and  these  real  variables  represent  but  one  kind  of  entity 
which  the  authors  have  chosen  to  call  elementary  propositions.  The 
most  general  statements  are  formed  by  merely  combining  these  variables 
by  means  of  the  two  primitive  propositional  functions  of  propositions 
Negation  and  Disjunction;  and  the  entire  theory  is  concerned  with  the 
process  of  asserting  those  combinations  which  it  regards  as  true  propositions, 
employing  for  this  purpose  a few  general  rules  which  tell  how  to  assert 
new  combinations  from  old,  and  a certain  number  of  primitive  assertions 
from  which  to  begin. 

This  theory  in  a somewhat  different  form  has  long  been  the  subject 
matter  of  symbolic  logic. J However,  although  it  had  reached  a high  state 
of  development  as  a theory  of  classes,  it  had  this  incurable  defect  as  a 
logic  of  propositions,  that  it  used  informally  in  its  proofs  the  very  proposi- 
tions whose  formal  statements  it  tried  to  prove.  This  defect  appears  to  be 
entirely  overcome  in  the  development  of  ‘ Principia/  But  owing  to  the 
particular  purpose  the  authors  had  in  view  they  decided  not  to  burden  their 
work  with  more  than  was  absolutely  necessary  for  its  achievement,  and  so 
gave  up  the  generality  of  outlook  which  characterized  symbolic  logic. 

It  is  with  the  recovery  of  this  generality  that  the  first  portion  of  our 
paper  deals.  We  here  wish  to  emphasize  that  the  theorems  of  this  paper 

* A.  N.  Whitehead  and  B.  Russell,  Principia  Mathematica,  Vol.  1,  1910;  Vol.  2 1912- 
Vol.  3,  1913.  Camb.  Univ.  Press. 

t Ibid.,  Vol.  1,  part  1,  section  A. 

t See  C.  I.  Lewis,  “A  Survey  of  Symbolic  Logic,”  University  of  California  Press,  1918. 
An  extensive  bibliography  is  given  there. 


1G4 


Post:  A General  Theory  of  Elementary  Propositions. 


are  about  the  logic  of  propositions  but  are  not  included  therein.  More 
particularly,  whereas  the  propositions  of  ‘Principia’  are  particular  asser-) 
tions  introduced  for  their  interest  and  usefulness  in  later  portions  of  the 
work,  those  of  the  present  paper  are  about  the  set  of  all  such  possible  , 
assertions.  Our  most  important  theorem  gives  a uniform  method  foii 
testing  the  truth  of  any  proposition  of  the  system;  and  by  means  of  this) 
theorem  it  becomes  possible  to  exhibit  certain  general  relations  which  exist 
between  these  propositions.  These  relations  definitely  show  that  the 
postulates  of  ‘ Principia  ’ are  capable  of  developing  the  complete  system  of 
the  logic  of  propositions  without  ever  introducing  results  extraneous  to  that; 
system — a conclusion  that  could  hardly  have  been  arrived  at  by  the  particun 
lar  processes  used  in  that  work. 

Further  development  suggests  itself  in  two  directions.  On  the  one  hand 
this  general  procedure  might  be  extended  to  other  portions  of  ‘ Principia, 
and  we  hope  at  some  future  time  to  present  the  beginning  of  such  an  at-, 
tempt.  On  the  other  hand  we  might  take  cognizance  of  the  fact  that  the. 
system  of  ‘Principia’  is  but  one  particular  development  of  the  theory — 
particular  in  the  primitive  functions  it  employs  and  in  the  postulates  it 
imposes  on  those  functions — and  so  might  construct  a general  theory  of  such 
developments.  This  we  have  tried  to  do  in  the  other  portions  of  the  paper. 
Our  first  generalization  leads  to  systems  which  are  essentially  equivalent 
to  that  of  ‘Principia’  and  connects  up  with  the  work  of  Sheffer*  and  Nicodj 
in  reducing  the  number  of  primitive  functions  and  of  primitive  propositions 
.respectively.  The  second  generalization,  on  the  other  hand,  while  including 
the  first  also  seems  to  introduce  essentially  new  systems.  One  class  of  such 
systems,  and  we  study  these  in  detail,  seems  to  have  the  same  relation  to 
ordinary  logic  that  geometry  in  a space  of  an  arbitrary  number  of  dimen- 
sions has  to  the  geometry  of  Euclid.  Whether  these  ‘‘ non- Aristotelian  ’ 
logics  and  the  general  development  which  includes  them  will  have  a direct 
application  we  do  not  know;  but  we  believe  that  inasmuch  as  the  theory 
of  elementary  propositions  is  at  the  base  of  the  complete  system  of  ‘ Prin- 
cipia,’ this  broadened  outlook  upon  the  theory  will  serve  to  prepare  us  foi 
a similar  analysis  of  that  complete  system,  and  so  ultimately  of  mathe- 
matics. 

Finally  a word  must  be  said  about  the  viewpoint  that  is  adopted  in  this 
paper  and  the  method  that  is  used.  We  have  consistently  regarded  the 
system  of  ‘ Principia  ’ and  the  generalizations  thereof  as  purely  formal  de- 

* H.  M.  Sheffer,  “A  Set  of  Five  Independent  Postulates  for  Boolean  Algebras,  with 
Applications  to  Logical  Constants,”  Trans . Amer.  Math.  Soc.,  14  (1913),  pp.  481-88. 

f J.  G.  P.  Nicod,  “A  Reduction  in  the  Number  of  the  Primitive  Propositions  of  Logic,” 
Proc.  Carnb.  Phil.  Soc.,  Vol.  XIX,  Jan.,  1917. 


■ 

J • 

Post:  A General  Theory  of  Elementary  Propositions.  165 

\velopments,*  and  so  have  used  whatever  instruments  of  logic  or  mathe- 
matics we  found  useful  for  a study  of  these  developments.  The  fact  that 
tone  of  the  interpretations  of  the  system  of  ‘Principia’  is  part  of  the 
informal  logic  we  have  used  in  this  study  makes  the  full  significance  of  this 
1 interpretation,  at  least  with  regard  to  proofs  of  consistency,  uncertain,  but 
it  in  no  way  affects  the  actual  content  of  the  paper  which  is  in  connection 
with  the  formal  systems. 

I welcome  this  opportunity  to  thank  Prof.  Keyser  for  the  many  sugges- 
tions he  has  offered  in  connection  with  this  paper,  as  well  as  for  the  labor 
he  assumed  in  reading  and  correcting  it. 

The  System  of  Pkincipia  Mathematica. 

1.  Description  of  the  System. — Let  p,  pi,  p2,  • • •,  q,  qi,  q2,  • • •,  r,  rit 
r2,  • • • arbitrarily  represent  the  variable  elementary  propositions  mentioned 
lin  the  introduction.  Then  by  means  of  the  two  primitive  functions  ~ p 
■ read  not  p — the  function  of  Negation)  and  p V q (p  or  q — the  function  of 
Disjunction)  with  the  aid  of  the  primitive  propositions 
I.  If  p is  an  elementary  proposition  ~ p is  an  elementary  proposition, 

If  p and  q are  elementary  propositions  p V q is  an  elementary  proposi- 
tion, 

we  combine  these  variables  to  form  the  various  propositions  or  rather 
ambiguous  values  of  propositional  functions  of  the  system.  It  is  desirable 
in  what  follows  to  have  before  us  the  vision  of  the  totality  of  these  functions 
streaming  out  from  the  unmodified  variable  p through  forms  of  ever  growing 
complexity  to  form  the  infinite  triangular  array 

V 

V V p,  Pi  V V2,  ~ P 

P V ~ p,  * * * > ~ Pi  V ~ P2,  • • (pi  V Pi)  V (Ps  V pf), 

~ (Pi  V Pi),  ~ (p  V p),  . ~ ^ p 


and  to  note  and  remember  that  this  array  of  functions  formed  merely 
through  combining  p’s  by  ^’s  and  V ’s  constitutes  the  entire  set  of  enuncia- 
tions it  is  possible  to  make  in  the  theory  of  elementary  propositions  of 
4 Principiad 

But  the  actual  theory  is  concerned  with  the  assertion  of  a certain  subset 
pf  these  functions.  We  denote  the  assertion  of  a function  by  writing  f- 
before  it.  Then  the  motive  power  for  the  resulting  process  of  deduction 
is  furnished  by  the  two  rules  of  operation: 

* For  a general  statement  of  this  viewpoint  see  C.  I.  Lewis,  Loc..  Cit.,  Chapter  VI, 
section  III. 


166 


Post:  A General  Theory  of  Elementary  Propositions. 


I 


i 

II.  The  assertion  of  a function  involving  a variable  p produces  the 
assertion  of  any  function  found  from  the  given  one  by  substituting  for  4 
any  other  variable  q,  or  ^ q,  or  (q  V r).* 

III.  “\-  P”  and  : ~~  P . V • Q ” produce  “\-  Q.” 

These  enable  us  to  assert  new  functions  from  old,  or  rather  in  the  form  in 
which  we  have  put  them,  generate  new  assertions  from  old.  And  the  com- 
plete set  of  assertions  is  produced  by  applying  II  and  III  both  to  the  follow- 
ing assertions  which  give  us  the  start,  and  to  all  derived  assertions  that  may 
result: 


IV.  b : ~ (p  V v)  • V • P,  h : ~ [_p  V {q  V . V • q V (p  V r), 

^ q . \/  . p \/  q,  b : . ~ q V r) . V •*  ^ (p  V q)  • V . p V 


b : ^ (p  V q)  . V . q V p. 


We  here  again  point  out  what  was  emphasized  in  the  introduction  tha  ) 
this  theory  concerns  itself  exclusively  with  the  production  of  particular 
assertions  through  the  detailed  use  of  the  rules  of  operation  upon  the 
primitive  assertions,  and  as  a consequence  the  set  of  theorems  of  this  portion 
of  ‘Principia’  consists  of  the  assertions  of  a certain  number  of  particulai 
functions  of  the  above  infinite  set.f 

2.  Truth-Table  Development^ — Let  us  denote  the  truth-value  of  an^ 
proposition  p by  + if  it  is  true  and  by  — if  it  is  false.  This  meaning  of  + 
and  — is  convenient  to  bear  in  mind  as  a guide  to  thought,  but  in  the' 
actual  development  that  follows  they  are  to  be  considered  merely  as  symbols 
which  we  manipulate  in  a certain  way.  Then  if  we  attach  these  two 
primitive  truth-tables  to  ~ and  V 


* This  operation  is  not  explicitly  stated  in  ‘ Principia  ’ but  is  pointed  out  to  be  neces- 
sary by  B.  Russell  in  his  “Introduction  to  Mathematical  Philosophy,”  London,  1919,  p.  151 . 
Its  particular  form  was  suggested  to  us  by  the  first  portion  of  the  operation  of  “Substiti. 
tion”  given  by  Lewis,  loc.  tit.,  p.  295.  It  will  be  noticed  that  the  effect  of  II  is  to  enable 
us  to  substitute  any  function  of  the  system  for  a variable  of  an  asserted  function. 

f We  have  consistently  ignored  the  idea  of  definition  in  this  description.  We  here 
rigorously  follow  the  authors  in  saying  that  definition  is  a convenience  but  not  a necessity 
and  so  need  not  be  considered  part  of  the  theoretical  development.  And  so  although  we 
too  shall  at  times  use  its  shorthand,  we  do  not  encumber  our  theoretical  survey  with  it. 

| Truth-values,  truth-functions  and  our  primitive  truth-tables  are  described  in  ‘ Prin- 
cipia,’ Vol.  1,  p.  8 and  p.  120,  but  the  general  notion  of  truth-table  is  not  introduced. 
This  notion  is  quite  precise  with  J’evons  and  Venn  (see  Lewis,  loc.  titus,  p.  74  and  pp.  175 
et  seq.  respectively)  and  has  its  foundation  in  the  formula  for  the  expansion  of  logical 
functions  first  given  by  Boole.  (G.  Boole,  “An  Investigation  of  the  Laws  of  Thought/ 
London,  Walton,  1854,  especially  pp.  72-76.)  For  the  relation  to  Schroder  see  the  foot- 
note to  section  3 


Post:  A General  Theory  of  Elementary  Propositions. 


167 


V 

~ v v,  q 

yv? 

1 + 

+ + 

+ 

_ 

+ + - 

+ 

t 

- + 

+ 

■ 

— — 

— 

We  have  a means  of  calculating  the  truth- values  of  ~ p and  p V q from  those 
of  their  arguments.  Now  consider  any  function  f(pi,  P2,  • • • pn)  in  our 
system  of  functions,  which  we  will  designate  by  F.  Then  since  / is  built  up 
of  combinations  of  ~’s  and  v’s,  if  we  assign  any  particular  set  of  truth- 
values  to  the  p’s,  successive  application  of  the  above  two  primitive  tables 
will  enable  us  to  calculate  the  corresponding  truth-value  of  f.  So  corre- 
sponding to  each  of  the  2n  possible  truth-configurations  of  the  p’s  a definite 
truth- value  of  / is  determined.  The  relation  thus  effected  we  shall  call  the 
truth-table  of/. 

For  example  consider  the  function 

~ ('v  V q)  V ~ ? V v)) 

which  is  the  ultimate  definition  of  the  function  p = q of  Principia.  We 
have  when  p is  + and  q is  + the  following  truth-values  of  the  successive 
Components  of  the  function  and  so  finally  of  the  function: 

p : +,  ~p\— , ~pVg:+,  ~ p V q)  '•  ~ 

q : +,  ~ q • ~ , ~ q V P ’ + , ~ q V p)  : ~ 

^ p v q)  v ~ q v v) : — t ~ p v q)  v ^ # v p))  • + 

the  successive  truth-values  being  found  by  direct  application  of  the  primitive 
tables.  In  the  same  way  the  truth-values  for  p +,  q — etc.  can  be  calcu- 
lated and  so  we  finally  get  the  truth-table  of  p = q,  i.e., 


v>  q 

p=  q 

+ + 

+ 

+ - 

— 

- + 

— 

— 

+ 

It  is  needless  to  say  that  in  actual  work  this  amount  of  detail  is  quite 
unnecessary. 

We  shall  call  the  number  of  variables  which  appear  in  a function  the 
order  of  that  function  as  well  as  that  of  its  truth-table.  It  is  evident  that 
there  are  22n  tables  of  the  nth  order.  We  now  prove  the 

Theorem.  To  every  truth-table  of  whatever  order  there  corresponds  at 
least  one  function  of  F which  has  it  for  its  truth-table. 


168 


Post:  A General  Theory  of  Elementary  Propositions. 


+ + 


For  first  corresponding  to  the  four  tables  of  the  first  order 
1 1 I,  1 1 + we  have  the  functions  p V P,  V V ~ p,  ~ (p  V ~ p),  ~ p-  No\ 
assume  there  is  a function  for  each  mth  order  table.  Then  in  any  table  c 
order  m + 1 the  configurations  for  which  pm+i  is  + constitute  an  mth  order 
table  for  which  there  is  some  function  fi(pi,  P2,  • • • pm )•  Likewise  cor- 
responding to  pm+i  — we  obtain /2(pi,  p2,  •••  pm).  Let  p.q  stand  fc? 
~ p V ~ q)  a function  which  has  the  truth-table 


v,  q 

p.q 

+ + 

+ 

+ - 

— 

- + 

— 

— — 

— 

Vm) 


Then  it  easily  follows  that  the  function 

Pm+ 1 -fl(Pl,  P2,  * * * Pm)  • V • ^ Pm+1  -fifa,  P2, 

has  for  its  truth-table  the  given  m + 1st  order  table. 

The  functions  of  F can  then  be  classified -according  to  their  tables  as 
follows:  those  which  have  all  their  truth- values  + , all  — , or  some  + and 
some  — . We  shall  call  these  functions  respectively  positive,  negative,  and 
mixed.  This  classification  is  of  great  importance  in  connection  with  the 
process  of  substitution  which  is  so  fundamental  in  the  postulational  de- 
velopment. We  shall  say  that  any  function  obtained  from  another  by  the 
process  of  substitution  is  contained  in  that  function.  We  then  have  the 
Theorem.  Every  function  contained  in  a positive  function  is  positive ; 
every  function  contained  in  a negative  function  is  negative ; every  mixed  function 
contains  at  least  one  function  for  every  possible  truth-table. 

The  first  two  results  are  immediate.  In  the  third  case  note  that  any 
mixed  function  f(pi,  p2,  • •-  • pn ) has  at  least  one  configuration  which  yields 
+ and  one  which  yields  — . Let  the  truth-value  of  pi  in  the  positive  con- 
figuration be  denoted  by  U and  in  the  negative  by  t[,  and  construct  a func- 
tion 0t(p)  with  the  truth-table 


V 

+ 

L 

— 

Then  \p(p)  = f(<f>i(p),  02 (p),  • • • <hn(p))  will  be  + when  p is  + and  — 
when  p is  — . But  by  our  first  theorem  there  is  at  least  one  function 
g(qi,  q2,  * * * qm)  corresponding  to  any  table  of  order  m.  Hence  ^p\jg{qi,  q2, 
' * * is  a function  contained  in  f(pi,  p2,  • • • pn)  corresponding  to  that 
table. 

Corollary.  Every  mixed  function  contains  at  least  one  positive  function 
and  one  negative  function. 


Post:  A General  Theory  of  Elementary  Propositions. 


169 


3.  The  Fundamental  Theorem.* — A necessary  and  sufficient  condition  that 
a function  of  F he  asserted  as  a result  of  the  postulates  II,  III,  IV  is  that  all 
its  truth-values  he  + . 

Note  first  that  each  of  the  primitive  assertions  of  IV  is  a positive  func- 
tion. Furthermore  from  the  assertion  of  positive  functions  we  can  only 
get  positive  functions.  For  the  only  method  we  have  of  producing  new 
assertions  from  old  is  through  the  use  of  II  and  III.  Now  II  can  only 
produce  positive  functions  since  every  function  contained  in  a positive 
function  is  positive.  As  for  III,  if  P is  + and  Q is  — , ~ P V Q is  — , so 
that  so  long  as  P is  a positive  function  and  ~ P V Q is  a positive  function  Q 
must  be  positive,  so  that  III  can  only  produce  positive  functions.  Hence 
every  asserted  function  is  positive  and  we  have  proved  the  condition 
necessary. 

In  order  to  prove  it  also  sufficient  we  give  a method  for  deriving  the 
assertion  of  any  positive  function.  It  will  simplify  the  exposition  to  intro- 
duce the  other  two  defined  functions  of  ‘Principia*  besides  p.q  (p  and  q) 
given  above,  viz., 

pDq.=  .~p\/q  Df  f;  p = q . = .p  3 q . q =>  p Df 

read  “p  implies  q”  and  “p  is  equivalent  to  q”  respectively,  and  having 
the  tables 


p,  q 

v =>  q p,q 

p=  q 

+ + 

+ + + 

+ 

+ - 

+ - 

— 

- + 

+ - + 

— 

— 

+ 

+ 

* The  method  for  testing  propositions  embodied  in  this  theorem  is  essentially  the  same 
as  that  given  by  Schroder  for  the  logical  system  he  has  developed.  (Ernst  Schroder,  Vor- 
lesungen  iiber  die  Algebra  Der  Logik,  Leipzig,  Teubner;  2.  Bd.  1.  Abth,  1891;  §32.)  But 
we  believe  the  range  of  significance  of  the  proof  we  have  given  to  be  quite  different  from 
that  of  the  work  of  Schroder.  For  first,  as  has  been  emphasized  by  Lewis  ( Loc . cit.,  Chap. 
IV),  formal  and  informal  logic  are  inextricably  bound  together  in  Schroder’s  development 
to  an  extent  that  prevents  the  system  as  a whole  from  being  completely  determined.  As 
a result  the  necessity  of  the  condition  of  the  theorem,  which  evidently  requires  such  a 
complete  determination  if  it  is  to  be  proved,  remains  unproved.  As  for  the  sufficiency, 
parts  E and  C of  our  proof  appear  in  the  proof  for  the  expression  of  functions  given  by 
Schroder.  (1.  Bd,  1890).  Part  A,  however,  seems  not  to  have  been  given  explicitly, 
while  corresponding  to  part  D are  all  the  theoretical  difficulties  met  with  in  passing  from 
the  theory  of  classes  to  that  of  propositions  when  the  development  is  not  strictly  formal. 
Hence  the  sufficiency  of  the  condition  is  only  incompletely  proved.  The  theorem  as  given 
by  Schroder  is  therefore  of  only  partial  significance  even  in  his  own  system;  and  when 
transplanted  to  the  system  of  Principia  requires  independent  proof.  Finally  we  may  men- 
tion that  the  applications  we  have  made  of  the  theorem  depend  for  their  significance  on 
those  parts  of  the  proof  which  do  not  appear,  and  could  not  appear,  in  Schroder, 
t III  can  now  be  written  “ |—  P”  and  “ H P D Q”  produce  “ t-  Q.” 


170 


Post:  A General  Theory  of  Elementary  Propositions. 


It  will  be  noticed  that  if  we  have  “ b fi(Vb  • • • pn)  — Ji(p  1,  • * • pn)  ” this 
asserted  equivalence  must  have  a positive  table  by  the  first  part  of  our) 
theorem,  and  so  fi  and  f2  must  have  the  same  truth-values  for  the  same 
configurations,  i.e.,  they  must  have  the  same  truth-table. 

The  proof  is  most  conveniently  given  in  four  stages. 

A.  We  prove  the  theorem  p = q .o  ./(p)  = f(q)  where  the  function  f 
may  involve  other  arguments  besides  the  one  indicated  and  need  not  involve 
that.  By  means  of  this  theorem  we  shall  be  able  to  replace  a constituent 
of  a given  function  by  any  equivalent  function,  and  have  the  result  equiva- 
lent to  the  given  function. 

It  becomes  necessary  for  the  first  time  to  introduce  the  notion  of  the 
rank  of  a function  which  we  define  inductively  as  follows:  the  unmodified 
variable  p will  be  said  to  be  of  rank  zero,  the  negative  of  a function  of  rank  rri. 
will  be  of  rank  m + 1 ; the  logical  sum  of  two  functions  the  rank  of  one  oi 
which  equals  and  the  other  does  not  exceed  m will  be  of  rank  m + 1.  Each 
function  of  F then  is  of  finite  rank  as  well  as  of  finite  order.*  Returning 
now  to  the  theorem  we  notice  that  it  is  true  for  a function  of  rank  zero 
since  it  reduces  either  to  p = g . d . p = q which  follows  from  pop]  by  II, 
or  to  p = q .o  .r  = r which  follows  from  p o . q d p,  r = r,  III  and  II 
Assume  now  that  the  theorem  holds  for  functions  of  rank  m and  lower. 
Then  it  also  holds  for  functions  of  rank  m + 1.  For  if  / is  of  rank  m + 1 
it  can  be  written  in  the  form  ~/i(p),  or,  /2(p)  V /3(p)  where  /i, /2  and  /3 
are  at  most  of  rank  m;  and  then  the  theorem  follows  by  using  p = q . o , 
~ p = ~ q,  p = q . o : . r = s : o : p V r . = . q V s along  with  p o q : o : 
q d r . d .p  o r,  III  and  II. 

B.  Consider  now  any  function  f(p1}  p2,  • • • pn )•  Using  ~ (p  V q) 
= . ~ p . ~~  q and  ~ ~ p = p with  the  aid  of  the  equivalence  theorem  of 
A and  p = q : o : q = r .o  .p  = r we  finally  obtain  f(p1}  p2,  • • • pn ) 
equivalent  to  a function /'(pi,  P2,  • • • pn ) which  is  expressed  merely  through 
combinations  of  p’s  and  ~ p’s  by  • ’s  and  V ’s. 

C. f  If  we  then  apply  the  distributive  law  of  logical  multiplication  to/', 
it  will  be  reduced  to  an  equivalent  function  consisting  of  successive  logica . 
sums  of  successive  logical  products  of  the  p’s  and  ~ p’s.  If  any  of  these 
products  has  neither  pn  nor  ~ pn  as  a factor  we  can  introduce  them  through 
the  propositions  p V ~ p,  and  p : o : q . = ..p  . q,  whence  q : = : (p  V ~ v) 
,g:  = :p-g.V.~p.g.  Now  apply  the  commutative  and  associative  laws 

* But  whereas  the  number  of  functions  of  given  order  is  infinite  those  of  given  rank 
are  finite. 

t This  as  well  as  all  other  particular  assertions  that  we  use  without  an  indication  oi 
proof  appear  in  Principia,  Vol.  I,  Part  A. 

J This  portion  of  the  proof  is  essentially  that  given  by  A.  N.  Whitehead  in  his  “Univer- 
sal Algebra,”  p.  46.  Camb.  Univ.  Press,  1898. 


It 


Post:  A General  Theory  of  Elementary  Propositions. 


171 


of  logical  multiplication  along  with  p .p  .=  .p  so  that  each  product  has  at 
most  one  pi  and  one  ~ pi.  Again  using  the  distributive  law  for  purposes 
of  factorization  along  with  the  commutative  and  associative  laws  of  addition 
we  finally  obtain  / equivalent  to 

fl(pi,  P2,---  Pn- 1)  • Pn  . ~ pn:  V :/2(pi,  • • • Pn-l)  • Pn  • V .fs(pi,  * * * Pn-l)  • ~ Pn 
where  one  or  more  of  the  terms  and  arguments  may  not  appear. 

D.  Suppose  now  that  the  original  function  is  positive;  then  this  equiva- 
lent function  will  be  positive.  If  in  particular  it  be  of  first  order,  it  can 
only  be  p V ~ p or  p . ~ .p  V .p  V ^ p.  The  first  is  an  asserted  function; 
likewise  the  second  through  p .d  . q V P-  Hence  also  f(p)  will  be  asserted 
through  p=  q.^-q  ^ p\  and  so  every  positive  first  order  function  is 
asserted.  Assume  now  that  this  is  true  for  all  mth  and  lower  ordered  func- 
tions and  let  / be  any  positive  (m  + l)st  order  function.  The  reduced 
function  being  then  positive,  both  /2  and  /3  will  be  positive,  and  hence  will 
be  asserted.  From  the  use  of  p : d : q . d .p  = q,  p .r  . V . p . ~ r : = : p 
. (r  V ~ r),  p : d : S . d .p  .S,  and  p . d . q V p,  the  reduced  function  will 
be  asserted  and  so  finally  /.  Hence  every  positive  function  can  be  asserted 
and  so  the  proof  is  complete. 

We  thus  see  that  given  any  function  the  theorem  gives  a direct  method 
for  testing  whether  that  function  can  or  cannot  be  asserted;  and  if  the  test 
shows  that  the  function  can  be  asserted  the  above  proof  will  give  us  an 
actual  method  for  immediately  writing  down  a formal  derivation  of  its  assertion 
by  means  of  the  postulates  of  Principia. 

Before  we  pass  on  to  theorems  about  the  system  itself  irrespective  of 
truth-tables  we  give  the  following  definitions  which  apply  directly  to  the 
system:  a true  function  is  one  that  can  be  asserted  as  a result  of  the  postu- 
lates, any  other  is  false;  a completely  false  function  is  a false  function  such 
that  every  function  therein  contained  is  false — otherwise  we  call  it  incom- 
pletely false.  We  then  have  the 

Corollary.  The  set  of  true,  completely  false,  and  incompletely  false 
functions  is  identical  with  the  set  of  positive,  negative,  and  mixed  functions 
respectively. 

4.  Consequences  of  the  Fundamental  Theorem. — In  the  above  develop- 
ment the  truth-values  +,  — were  arbitrary  symbols  which  were  found 
related  in  certain  suggestive  ways  through  the  fundamental  theorem.  We 
are  now  in  a psoition  to  give  direct  definitions  of  these  truth- values  in  terms 
of  the  postulational  development.  In  fact  we  shall  define  + to  be  the  set 
of  true  functions,  — the  set  of  completely  false  functions.  The  truth- value 
of  a function  will  then  exist  when  and  only  when  it  is  true  or  completely 
false,  and  it  will  be  defined  as  that  class  (+,  — ) of  which  it  is  a member. 
The  content  of  the  fundamental  theorem  consists  now  of  these  two  theorems : 


172  Post:  A General  Theory  of  Elementary  Propositions. 

/ 

1 . The  truth- value  of  ~ p and  q\/  r exists  whenever  the  truth- values  of 
p,  q and  r exist,  and  depends  only  on  those  truth-values  as  given  by  the 
primitive  tables.  It  therefore  follows  that  the  same  is  true  of  any  function 
of  F,  and  that  the  truth-table  of  such  a function  can  be  directly  calculated 
from  the- primitive  tables. 

2.  The  fundamental  theorem  as  stated,  or  else  in  the  form:  if/i  and  jV 
is  any  pair  of  positive  and  negative  functions  respectively,  then  a necessary 
and  sufficient  condition  that  a function  f(pi,  P2,  • • • pn)  be  asserted  is  that, 
each  of  the  2n  contained  functions  found  by  substituting  fi  and  /2  for  the 
p’s  is  asserted.  It  will  be  noticed  that  theorem  (1)  tells  us  how  to  deter- 
mine whether  these  latter  are  asserted. 

We  now  pass  on  to  several  theorems  about  the  system. 

Theorem.  It  is  possible  to  find  22™  functions  of  order  n such  that  no  two 
of  them  are  equivalent  and  such  that  every  other  function  of  order  n is  equivalent 
to  one  of  these. 

For  we  can  find  22”  functions  corresponding  to  the  22n  different  tables  of 
order  n.  The  equivalence  of  any  two  of  these  will  then  not  have  a positive 
table  and  so  will  not  be  asserted.  On  the  other  hand  any  other  nth.  order 
function  will  have  the  same  table  as  one  of  the  22n  possible  tables,  and  so 
the  corresponding  equivalence  will  be  positive  and  hence  asserted. 

Theorem.  An  incompletely  false  function  contains  at  least  one  function 
for  each  given  function  which  is  equivalent  to  that  given  function. 

Corollary.  An  incompletely  false  function  contains  at  least  one  true 
function  and  one  completely  false  function. 

Theorem.  The  negative  of  a completely  false  function  is  true. 

For  a completely  false  function  has  a negative  truth-table,  and  so  its 
negative  will  have  a positive  table  and  hence  be  asserted.  It  is  worth 
noticing  that  although  this  theorem  is  immediate  once  we  have  the  funda- 
mental theorem  it  would  be  quite  difficult  without  it. 

Corollary.  Every  function  of  F is  either  true,  or  its  negative  is  true,  or 
it  contains  both  a true  function  and  one  whose  negative  is  true. 

Theorem.  The  system  of  elementary  propositions  of  ( Principia’  is 
consistent. 

For  if  it  were  inconsistent  we  would  have  both  a function  and  its  negative 
asserted.  But  then  both  the  function  and  its  negative  would  have  to 
have  positive  tables  whereas  if  a function  has  a positive  table  its  negative 
has  a negative  table.* 

Theorem.  Every  function  of  the  system  can  either  be  asserted  by  means 
of  the  postulates  or  else  is  inconsistent  with  them. 

* This  argument  requires  merely  the  first  part  of  the  fundamental  theorem  which 
was  proved  quite  simply. 


Post:  A General  Theory  of  Elementary  Propositions. 


173 


For  if  a function  be  not  asserted  as  a result  of  the  postulates  it  will 
contain  a function  whose  negative  can  be  so  asserted.  If  then  we  assert 
the  original  function,  the  contained  function  will  be  asserted  so  that  we 
have  asserted  both  a function  and  its  negative,  i.e.,  we  have  a contra- 
diction. 

Corollary.  A function  is  either  asserted  as  a result  of  the  postulates  or 
else  its  assertion  will  bring  about  the  assertion  of  every  possible  elementary 
proposition. 

For  by  the  theorem  we  would  obtain  the  assertion  of  both  a function 
and  its  negative  and  so  by  ~ p .d  .p  3 q the  assertion  of  the  unmodified 
variable  q.  But  q then  represents  any  elementary  proposition. 

In  conclusion  let  us  note  that  while  the  fundamental  theorem  shows 
that  the  postulates  bring  about  the  assertion  of  those  and  only  those 
theorems  which  should  belong  to  the  system,  this  last  theorem  enables  us 
to  say  that  they  also  automatically  exclude  the  very  possibility  of  any 
added  assertions. 

Generalization  by  Truth- Tables. 

5.  General  Survey  of  the  Systems  Generated. — The  system  we  have 
studied  in  the  preceding  sections  is  a particular  system  depending  upon  the 
two  primitive  functions  ~ p and  p V q.  Two  modes  of  attack  have  pre- 
sented themselves.  On  the  one  hand  we  have  the  original  postulational 
method,  on  the  other  the  truth-table  development.  In  passing  to  a general 
study  of  systems  of  the  kind  discussed  these  two  methods  present  themselves 
as  instruments  of  generalization.  We  reserve  the  postulational  generaliza- 
tion for  the  next  portion  of  our  paper  and  now  take  up  the  truth-table 
generalization. 

To  gain  complete  generality  let  us  assume  for  our  primitives  p arbitrary 
functions  with  an  arbitrary  number  of  arguments  which  we  will  designate  by 

fl(Pl,  P2,  * ‘ * Pm1),f2(Pl,  P2y  ‘ ‘ * Pm2),  ■*  * * (pi,  £2,  * * * Pm J 

and  let  us  attach  an  arbitrary  truth-table  to  each.  By  successive  combina- 
tions of  these  functions  with  different  or  repeated  arguments  we  generate 
the  set  of  derived  functions  which  as  before  we  designate  by  F.  Again 
each  function  of  F will  possess  a truth-table  in  virtue  of  the  tables  of  the 
primitive  functions  of  which  each  is  composed.  Denote  the  set  of  truth- 
tables  thus  generated  by  T.  Then  whereas  in  the  system  of  ‘ Principia 5 T 
consists  of  all  possible  truth-tables,  this  will  not  necessarily  be  the  case  here. 

In  another  paper  we  completely  determine  all  the  possible  systems  T 
and  show  that  there  are  66  systems  that  can  be  generated  by  tables  of  third  and 
lower  order,  and  8 infinite  families  of  systems  that  are  generated  by  the  intro- 
duction of  fourth  and  higher  ordered  tables. 


174 


Post:  A General  Theory  of  Elementary  Propositions. 


If  two  systems  have  the  same  truth-tables  the  primitives  of  each  can 
evidently  be  expressed  in  terms  of  those  of  the  other  so  that  truth-tables 
are  preserved.  We  can  then  say  that  each  system  has  a representation  in 
the  other  and  the  two  are  equivalent.  In  particular  every  truth-system  has  a 
representation  in  the  system  of  Principia  while  every  complete  system , i.e., 
having  all  possible  truth-tables,  is  equivalent  to  it.  In  the  aforementioned 
paper  we  also  determine  the  ways  in  which  a complete  system  may  be 
. generated,  and  it  turns  out  that  one  table  alone  is  sufficient  to  generate  it, 
and  it  can  be  either  of  these  two 


+ + 

— 

+ + 

— 

+ - 

+ 

+ - 

— 

- + 

+ 

- + 

— 

— 

+ 

— 

+ 

a result  first  given  by  Sheffer  as  stated  in  the  introduction. 

The  truth-table  development  for  complete  systems  is  essentially  the 
same  as  that  given  in  section  2.  It  is  easy  to  prove  for  all  systems  the 
Theorem.  Every  function  contained  in  a positive  function  is  positive ; 
every  function  contained  in  a negative  function  is  negative ; every  mixed  function 
contains  a function  for  every  table  of  the  system. 

6.  Postulates  for  a Complete  System. — We  now  show  how  to  construct 
a set  of  postulates  for  any  complete  system  such  that:  the  set  of  asserted 
functions  is  identical  with  the  set  of  positive  functions,  while  the  assertion  of  any 
other  function  brings  about  the  assertion  of  every  elementary  proposition 
a property  which  also  characterized  the  system  of  ‘ Principia.’ 

Let  ^'p  and  p V ' q be  functions  in  the  given  complete  system  with  the 
tables  of  ~ and  V.  Out  of  and  V'  we  then  construct  p d'  q and 
p = ' q as  p d q and  p = q are  found  from  ~ and  V , and  also/1  (pi,  • • • pmi), 
* ' ‘>fUPb  * * * £W)  with  the  same  tables  as/i(p,  • • • pmi),  • • -,fp(pi,  * * • PmJ- 
This  is  possible  since  ~ and  V , and  so  and  V ' can  generate  a complete 
system.  All  the  functions  V',  d ',  =',  /l,  • • • fl  are  ultimately  ex- 
pressed in  terms  of  the/’s  and  so  belong  to  the  system.  Construct  now  the 
following  set  of  postulates: 

I.  If  pi,  • • • pmi  are  elementary  propositions,  fi(pi,  • • • pmi)  is. 

If  pi,  • • • pmfJL  are  elementary  propositions,  f^(pi,  • • • pm M)  is. 

II.  The  assertion  of  a function  involving  a variable  p produces  the  assertion 
of  any  function  found  from  the  given  one  by  substituting  for  p any  other 
variable  q,  or/i(gi,  • • • qmi),  • • • or/M(gi,  • 

III.  “HP”  and  “bP  a'Q”  produces  “ b Q” 

IV.  (1)  1-  : p V ' p . 3 ' p (a)  h P2,  ■■■  Pm,)  = 'f(Ph  Pi>  ■■■  Pm,), 


(5)  h • • • 


O)  I-  -U(Pl,  Pi,  ■ ■ • Pm*)  ='fl(Pl>  Pi,  ■■■  Pm m). 


Post:  A General  Theory  of  Elementary  Propositions.  175 

where  (l)-(5)  are  the  assertions  of  IV  in  sec.  1 with  and  V'  in  place  of 
~ and  V . 

That  all  asserted  functions  are  positive  can  be  verified  as  in  the  proof  of 
sec.  4.  As  for  the  converse,  note  that  III  and  IV  (l)-(5)  being  of  the  same 
form  as  III  and  IV  of  sec.  4 will  yield  the  assertion  of  all  positive  functions 
expressed  in  terms  of  and  V'.  By  the  use  of  (a)-(u)  every  function 
can  be  shown  to  be  equivalent  (=')  to  some  function  expressed  by 
and  V ' and  so  every  positive  function  will  be  asserted.  In  the  same  way 
the  assertion  of  any  non-positive  function  will  bring  about  the  assertion  of 
a non-positive  function  in  and  V'  alone,  and  so  of  any  proposition. 

We  thus  see  that  complete  systems  are  equivalent  to  the  system  of 
‘Principia’  not  only  in  the  truth  table  development  but  also  postulationally. 
As  other  systems  are  in  a sense  degenerate  forms  of  complete  systems  we 
can  conclude  that  no  new  logical  systems  are  introduced. 

7.  Application  to  Nicod’s  Postulate  Set. — Although,  as  in  most  existence 
theorems,  the  above  set  of  postulates  may  not  be  the  simplest  in  any  one 
case,  it  can  be  used  to  advantage  in  showing  that  a given  set  has  the  same 
property  as  it  possesses.  For  this  purpose  we  show  directly  that  all  asserted 
functions  are  positive,  and  then  that  by  means  of  the  given  postulates  (a) 
each  of  our  formal  postulates  may  be  derived  (6)  that  the  results  derivable 
by  our  informal  postulates  can  also  be  derived  by  the  given  ones.* 

As  an  example  we  consider  the  set  of  postulates  given  by  Nicod  for  the 
theory  of  elementary  propositions  in  terms  of  the  single  primitive  function 
of  Sheffer’s  which  Nicod  denotes  by  p\q  and  is  termed  incompatibility  by 
Russell. f It  is  the  first  of  the  two  functions  given  in  section  5 as  generating 
a complete  system.  Nicod  gives  the  definitions 

~p.  = .p\p  Df,  p V q.  = .p/p\qlq  Df 

which  we  take  to  be  our  ^'p  and  p V 'q  respectively.  His  p d q .=  . p | q/qDf 
however  is  not  our  p o'q  which  is  V 'q.  The  primary  distinction  of 
his  system  is  that  he  uses  but  one  formal  primitive  proposition. 

In  carrying  out  the  proof  suggested  we  merely  note  that  by  means  of  his 
informal  proposition  “\-  P”  and  “ b P|P/Q”  produce  “b  Q”  we  get  the 
effect  of  “bP”  and  “bP|Q/Q”  i.e.,  “b  P s Q”  produce  “b  Q”  when 
R = Q.  Since  he  has  p d ' g . d . p d q we  thus  get  the  effect  of  “ b P” 

* That  the  informal  postulates  of  a system  must  be  proved  effectively  replaced  by 
others  in  another  system  is  a precaution  rarely  taken  in  discussions  of  equivalence  or 
dependence  of  logical  systems.  Such  a discussion  is  unnecessary  in  ordinary  mathematical 
systems  since  their  distinctive  postulates  are  all  formal,  the  informal  ones  being  those  of 
a common  logic.  But  in  comparing  logical  systems,  which  usually  do  contain  different 
informal  postulates,  such  a discussion  is  fundamental. 

t B.  Russell,  loc.  cit.,  chap.  XIV. 


176  Post:  A General  Theory  of  Elementary  Propositions. 

and  “b  P s'  Q”  produce  “b  Q”  our  III.  Likewise  each  function  IV  is 
proved  with  however  d in  place  of  d But  by  means  of  p d q . d .p  d ' q 
this  too  is  remedied.  We  then  easily  complete  the  proof  of  the 
Theorem.  If  in  Nicod’s  system  we  give  to  p\q  the  table 


V , (1 

v \q 

+ + 

— 

+ - 

+ 

- + 

+ 

— 

+ 

then  the  set  of  asserted  functions  is  identical  with  the  resulting  set  of  positive 
functions',  and  the  assertion  of  any  other  function  would  bring  about  the  asser- 
tion of  every  elementary  proposition. 


Generalization  by  Postulation. 

8.  The  Generalized  Set  of  Postulates. — As  in  the  truth-table  develop- 
ment we  assume  arbitrary  primitive  functions  of  propositions 

fl(PU  P2,  Pm ,),  ‘ P2,  * ‘ * 2W); 

but  in  place  of  the  arbitrary  associated  truth-tables  we  have  a set  of  postu- 
lates of  the  following  form.  We  have  tried  to  preserve  all  the  informal 
properties  of  the  postulates  of  ‘Principia’  (and  of  sec.  5)  but  generalize 
the  formal  properties  completely. 

I.  (As  in  sec.  5.) 

II.  (As  in  sec.  5.) 

HI.  “ b gu(Pi,  Pi,  • • • Pkl)  ” • • • “ b gK1(Pi,  P2,  • • • P* J ” 


“b  gufP,  P2'"Pky 

produce 

"b^Pi,  p2,  •••  Pkjk 


“b^(Pi, Pi,  •••  PO” 

produce 

“\-g*{Pu p2,  •••  POM 


where  the  P’s  are  any  combinations  of  f s including  the  special  case  of  the 
unmodified  variable,  while  the  g’s  are  particular  combinations  of  this  kind 
which  need  not  have  all  the  indicated  arguments. 

IV.  b hi(pi,  p2,  " ' Ph) 

b h2(pi,  Pi,  • • • Ph) 


•b  hk($i,  Pi,  • • • pif) 

where  the  ^s  are  particular  combinations  of  the  f s. 

The  retention  of  I and  II  which  are  characteristic  of  the  theory  of 


Post:  A General  Theory  of  Elementary  Propositions. 


177 


elementary  propositions  is  our  justification  for  giving  that  name  to  the 
systems  that  may  be  generated  by  the  above  set  of  postulates.  In  what 
follows  we  give  what  we  consider  to  be  merely  an  introduction  to  the  general 
theory. 

9.  Definition  of  Consistency  and  Related  Concepts. — The  prime  requi- 
site of  a set  of  postulates  is  that  it  be  consistent.  Since  the  ordinary  notion 
of  consistency  involves  that  of  contradiction  which  again  involves  negation, 
and  since  this  function  does  not  appear  in  general  as  a primitive  in  the 
above  system  a new  definition  must  be  given. 

Now  an  inconsistent  system  in  the  ordinary  sense  will  involve  the  asser- 
tion of  a pair  of  contradictory  propositions  which  as  we  have  seen  will 
bring  about  the  assertion  of  every  elementary  proposition  through  the 
assertion  of  the  unmodified  variable  p.  Conversely  since  p stands  for  any 
elementary  proposition  its  assertion  would  yield  the  assertion  of  contra- 
dictory propositions  and  so  render  the  system  inconsistent.  The  two  notions 
are  thus  equivalent  in  ordinary  systems;  and  since  one  retains  significance 
in  the  general  case  we  are  led  to  the 

Definition. — A system  will  he  said  to  he  inconsistent  if  it  yields  the  asser- 
tion of  the  unmodified  variable  p. 

In  a consistent  system  we  may  then  define  a true  function  as  one  that 
can  be  asserted  as  a result  of  the  postulates.  Instead  of  defining  a false 
function  as  one  not  true,  we  give  the  following 

Definition.  A false  function  is  one  such  that  if  its  assertion  he  added 
to  the  postulates  the  system  is  rendered  inconsistent. 

We  can  then  state  that  in  the  system  of  ‘Principia’  every  function  is 
true  or  false.  This  suggests  the 

Definition.  If  every  function  of  a consistent  system  is  true  or  false  the 
system  will  he  said  to  he  closed  A 

As  a justification  of  this  name  we  may  note  that  the  postulates  of  such  a 
system  automatically  exclude  the  possibility  of  any  added  assertions — a 
state  of  affairs  we  believe  to  be  highly  desirable  in  the  final  form  of  a logical 
theory. 

10.  Properties  of  Consistent  Systems. — In  all  that  follows  we  assume 
that  the  system  discussed  is  consistent.  If  it  be  inconsistent  one  could 
hardly  say  anything  more  about  it. 

We  turn  to  a theorem  which  will  give  us  most  of  the  results  of  this 
section.  But  first  we  must  state  two  lemmas  which  we  do  not  further  prove. 

Lemma  1. — If  a given  set  of  functions  gives  rise  to  some  other  function 
in  accordance  with  II  and  III,  and  if  these  functions  involve  certain  letters 

* Had  the  name  not  been  in  use  in  a different  connection  we  should  have  introduced 
the  term  categorical. 


178  Post:  A General  Theory  of  Elementary  Propositions. 

r i,  7*2,  • • • ri  upon  which  no  substitution  is  made  in  the  process,  then  the 
same  deductive  process  will  be  valid  if  we  have  given  the  original  functions 
with  an  arbitrary  substitution  of  the  r’s  as  described  in  II  provided  this 
substitution  is  also  made  throughout  the  process. 

Lemma  2. — The  most  general  process  of  obtaining  an  assertion  from  a 
given  set  of  assertions  in  accordance  with  II  and  III  can  be  reduced  to 
first  asserting  a number  of  functions  in  accordance  with  II,  and  then  applying 
II  and  III  in  such  a way  that  no  substitutions  are  made  on  the  arguments  of 
those  functions. 

Theorem.  Every  false  function  contains  a finite  set  of  untrue  first  order 
functions  d>i{p),  02  (p),  * * * <f>v(v)  su°h  that  whenever  p is  replaced  by  an  untrue 
function  at  least  one  of  these  functions  remains  untrue. 

By  the  definition  of  false  functions  there  must  be  some  deductive  process 
whereby  from  the  given  false  function  and  true  functions  we  assert  p.  By 
lemma  2 we  can  replace  this  process  by  another  where  from  the  given  false 
function  and  true  functions  we  obtain  certain  contained  functions  from 
which  without  substitution  of  the  arguments  we  obtain  p.  Now  first  by 
lemma  1 we  can  equate  to  p all  the  arguments  thus  appearing  and  still 
have  a valid  deductive  process  for  obtaining  p.  Denote  the  resulting  untrue 
functions  which  are  contained  in  the  original  false  function  by  0i(p), 
</>2 (p),  • • • 0„(p)*  Then  secondly  by  lemma  1 we  can  replace  p by  any 
function  \p  and  still  have  a valid  process  which  now  consists  in  obtaining  0 
from  certain  true  functions  and  0i(0),  • • • 0„(0).  If  then  each  were 

true,  0,  being  obtained  from  true  functions  in  accordance  with  II  and  III 
would  be  true.  It  follows  that  if  0 be  untrue,  some  <^(0)  must  be  untrue. 

Theorem.  Every  false  function  contains  an  infinite  number  of  untrue 
first  order  functions)  and  if  the  system  has  at  least  one  false  function  of  order 
greater  than  one,  then  each  false  function  contains  an  infinite  number  of  untrue 
functions  of  every  order. 

By  the  above  theorem  the  false  function  contains  at  least  one  untrue 
function  0tl(p)-  By  the  same  theorem  some  </>ti0tl(p)  must  be  untrue,  etc., 
through  </>ly,  <Ny  l • • • 0tl(p).  These  are  all  different  being  of  different  rank, 
and  are  all  contained  in  the  given  function. 

The  last  part  of  the  theorem  may  then  be  proved  by  showing  that  by 
replacing  equal  by  unequal  variables  in  the  infinity  of  functions  thus 
gotten  from  the  false  function  of  order  greater  than  one  we  get  untrue 
functions  of  every  order,  and  so  by  the  above  method  an  infinite  number  of 
every  order  in  every  false  function. 

We  have  immediately  the 

Theorem.  A necessary  and  sufficient  condition  that  a function  of  a 
closed  system  be  true  is  that  all  contained  first  order  functions  be  true. 


Post:  A General  Theory  of  Elementary  Propositions. 


179 


Corollary.  It  is  also  necessary  and  sufficient  that  all  those  of  rank  greater 
than  some  finite  integer  p he  true. 

In  analogy  with  corresponding  ideas  in  the  system  of  ‘Principia’  de- 
fine a completely  untrue  function  as  one  in  which  all  contained  functions 
are  untrue  with  a similar  definition  for  completely  false.  We  then  have 
the  interesting 

Theorem.  If  a system  has  a completely  untrue  function,  then  every  false 
function  contains  a completely  untrue  function. 

Every  function  contained  in  the  completely  untrue  function  makes 
at  least  one  <f>t(p)  of  a false  function  untrue.  If  0 is  such  a contained  func- 
tion which  makes  say  0tl  (p)  true,  then  0 will  be  completely  untrue,  and 
all  contained  functions  will  make  0tl  (p)  true  yet  some  remaining  0t(p) 
untrue.  By  repeating  this  process  we  finally  obtain  a function  0'  such 
that  all  contained  functions  make  each  <f>t(p)  °f  a set  that  remains  untrue. 
Each  such  </>,  (0')  will  then  be  a completely  untrue  function  in  the  given  one. 

Corollary.  If  a closed  system  has  a completely  false  function  every 
false  function  contains  a completely  false  function. 

If  we  call  such  a system  completely  closed  we  have  the  stronger 

Theorem.  In  a completely  closed  system  every  false  function  f(pi,  P2, 
•••,  pn)  contains  a completely  false  function  /(0i(p),  02 (p),  •••,  0n(p)) 
where  ^each  xpfp)  is  either  true  or  completely  false. 

By  equating  all  variables  to  p in  the  function  of  the  corollary  we  get  such 
a completely  false  function  where  some  0’s  may  be  incompletely  false. 
These  are  then  eliminated  by  successively  substituting  for  p functions 
which  make  them  true. 

Corollary.  A necessary  and  sufficient  condition  that  a function  of  a 
completely  closed  system  he  true  is  that  all  contained  first  order  functions  found 
by  substituting  true  or  completely  false  functions  for  the  arguments  he  true. 

This  property  begins  to  approximate  to  the  truth-table  method.  It 
leads  us  easily  to  the  following  criterion  for  a completely  closed  postulational 
system  being  a truth-system  which  we  state  without  proof. 

Theorem.  A necessary  and  sufficient  condition  that  a completely  closed 
postulational  system  he  a truth-system  is  that  a true  first  order  function  remains 
true  whenever  we  replace  a true  or  completely  false  constituent  function  hy  any 
other  true  or  completely  false  first  order  function  respectively  A 

* In  making  a more  complete  study  of  the  postulational  generalization  it  would  be 
desirable  to  classify  all  the  systems  that  may  result  more  or  less  in  the  way  in  which  we 
have  classified  truth-systems  through  the  associated  systems  of  truth-tables.  In  this 
connection  we  might  define  the  order  of  a set  of  postulates  as  the  largest  number  of  premises 
used  in  deriving  a conclusion  in  III,  and  the  order  of  a system  as  the  lowest  order  a set  of 
postulates  deriving  it  can  have.  It  is  then  of  interest  to  note  that  whereas  the  set  of  postu- 
lates of  the  system  of  ‘Principia’  is  of  the  second  order , the  system  itself  is  of  the  first  order. 


180  Post:  A General  Theory  of  Elementary  Propositions. 

m- valued  Truth-Systems.* 

11.  The  Generalized  (~,  V)  System. — We  have  seen  that  the  truth- 
table  generalization,  at  least  with  regard  to  complete  systems,  is  included  in 
the  postulational  development.  We  now  show  that  the  latter  is  more 
general  by  presenting  a new  class  of  systems,  distinct  from  the  two-valued 
systems  of  symbolic  logic,  which  can  be  generated  by  a completely  closed 
set  of  postulates. 

In  these  systems  instead  of  the  two  truth-values  + , — we  have  m 
distinct  “truth-values”  ti,  t2,  • • *,  tm  where  m is  any  positive  integer.  A 
function  of  order  n will  now  have  mn  configurations  in  its  truth-table,  so 
that  there  will  be  mmn  truth-tables  of  order  n.  Calling  a system  having  all 
possible  tables  complete,  we  now  show  that  the  following  two  tables  generate 
a complete  system. 


n - jfi 
^2  — jt 


We  see  that  ~ mp , the  generalization  of  ~ p,  permutes  the  truth- values 
cyclically,  while  p V mq,  the  generalization  of  p V q has  the  higher  of  the 
two  truth-values.f 

To  construct  a function  for  any  first  order  table,  of  which  there  are  mm< 
note  that  , 

h (?)•  = • V • V m~lp : V » • • • ~ZrlV  Df, 

where  ~2p . = . p Df,  etc.,  has  all  its  truth  values  t\.  Then 

rmi(p):  = . .V«.p):V«.  Df 

1 — 

has  all  values  tm  except  the  first  which  is  tmi.  Any  first  order  table 


V 

f(p) 

tl 

tm. 

t2 

tm  2 

tm 

can  then  be  constructed  by  the  function 


v , q 

P V m^[ 

ht  i 

ti 

tji, 

If-fj  2 

tji 

tmtn 

tm 

V 

~ mP 

tl 

t2 

i'L 

U 

tm 

tl 

* See  Lewis,  loc.  tit.,  p.  222  for  the  term  “Two-Valued  Algebra.” 
f The  higher  truth- value  has  here  the  smaller  subscript. 


Post:  A General  Theory  of  Elementary  Propositions.  181 

T'mj(p)  • Vm  • p)  • V m • m p)  . . V m ^’mrtf^'mP)  • 

Construct  now  a function  for  the  table 


V 

^"mP 

tl 

tm 

h 

tjn—l 

tm 

tl 

and  define  p ,mq.=  . ~m(~mP  . Vm  . 2)/  which  is  the  generalization 

of  p .q  and  has  the  lower  of  the  two  truth  values  of  its  arguments.  We 
can  now  construct  a table  all  of  whose  values  are  tm  except  for  one  con- 
figuration tmv  tm2,  • • • ,tmn  when  it  is  ^ by  the  function 

TM(-rmi+,pL)  .mT^Z-mi+lp2)  . m * * * 

and  so  any  table  by  constructing  such  a function  for  each  configuration 
and  then  “ summing  up”  by  Vm* 

12.  Classification  of  Functions — the  m dimensional  Space  Analogy. — 

The  generalization  of  the  classification  of  functions  into  positive,  negative 
and  mixed  is  afforded  us  by  the  following 

Theorem.  A function  contains  at  least  one  function  for  every  truth-table 
whose  values  are  contained  among  the  values  of  the  given  table. 

Let  tm i • • • tm M be  the  truth-values  that  appear  in  the  table  of  a given 
function  f(pi,  P2,  • • • pn )•  Then  we  can  pick  out  p configurations  having 
these  values  respectively.  Construct  functions  (f>Ap ) such  that  when  p 
has  the  vlaue  tm.  of  one  of  these  configurations,  have  the  value  of  pi 

in  that  configuration.  It  is  then  easily  seen  that  f(4>i(p),  * * *,  4>n(p))  has 
the  value  tm.  whenever  p has  the  value  tm..  If  then  yp(qi,  qz,  • • *,  qi)  have 
a table  whose  values  are  among  the  tm’ s,  f(<f> i(^),-  • • *,  will  be  a 

function  contained  in  the  given  function  with  that  table. 

We  are  thus  led  to  a classification  of  functions  by  means  of  their  truth- 
tables  such  that  the  set  of  tables  pf  contained  in  a given  function  is  the  same 
for  all  functions  in  a given  class.  We  then  have  m classes  of  functions  where 
but  one  truth- value  appears,  | ~m(m  — l)]/2!  with  two  truth-values,  •••, 
\jn(m  — 1)  •••  (m—  p l)]/jul  with  p truth- values,  • • *,  one  class  with 
all  m truth-values.  We  thus  have  2m  — 1 classes  of  functions  which  when 
m — 2 reduces  to  the  three  classes  of  positive,  negative  and  mixed  functions. 

These  formulse  suggest  an  analogy  which,  if  well  founded,  is  of  great 
interest.  For  this  purpose  replace  the  set  of  functions  having  all  of  a given 
set  of  p truth-values  by  all  functions  whose  values  are  among  these  p values. 
If  then  we  compare  the  functions  of  our  complete  system  to  the  points  of  a 


182 


Post:  A General  Theory  of  Elementary  Propositions. 


space  of  m dimensions,*  the  m classes  of  functions  with  but  one  truth-value 
would  correspond  to  the  m coordinate  axes,  the  \jn(m  — l)]/2!  classes  of 
functions  with  no  more  than  two  truth- values  to  the  [m{m  — l)J/2!  co- 
ordinate planes,  etc.,  so  that  except  for  the  absence  of  an  origin  all  properties 
of  determination  and  intersection  within  the  coordinate  configurations  go 
over.  If  then  we  attach  the  name  m-dimensional  truth-space  to  our  system, 
we  observe  the  following  difference,  that  whereas  the  highest  dimensioned 
intuitional  point  space  is  three,  the  highest  dimensioned  intuitional  proposi- 
tion space  is  two.  But  just  as  we  can  interpret  the  higher  dimensioned 
spaces  of  geometry  intuitionally  by  using  some  other  element  than  point, 
so  we  shall  later  interpret  the  higher  dimensioned  spaces  of  our  logic  by 
taking  some  other  element  than  proposition. 

13.  Truth-Table  Characteristics  of  Asserted  Functions. — The  following 
analysis  presupposes  that  in  constructing  a set  of  postulates  for  the  system 
we  at  least  wish  to  impose  the 

Condition. — If  a function  is  asserted,  all  functions  with  the  same  truth- 
table  will  be  asserted. 

It  follows  from  the  theorem  of  the  preceding  section  that  under  the  given 
condition,  if  a function  is  asserted,  every  function  of  the  truth-space  it  determines 
is  asserted. 

We  can  now  prove  that  if  the  system  is  to  be  completely  closed  its  asserted 
functions  must  constitute  a single  truth-space  contained  in  the  given  truth  space. 
For  if  there  were  at  least  two  such  spaces,  then  a function  having  all  their 
truth- values  would  be  false,  and  so  would  contain  a completely  false  function. 
This  in  turn  would  contain  functions  with  but  one  truth- value;  and  these 
being  therefore  in  one  of  the  two  given  spaces  would  be  true  which  contra- 
dicts their  being  in  a completely  false  function. 

No  loss  of  generality  ensues  if  we  take  the  truth  values  of  this  contained 
truth-space  of  asserted  functions  to  be  t\,  t2,  • • • t^,  where,  to  avoid  degener- 
ate cases  0 < p < m.  We  now  show  that  a completely  closed  set  of  postu- 
lates can  be  constructed  for  all  such  systems. 

14.  A Completely  Closed  Set  of  Postulates  for  the  Systems. — I and  II 
are  determined  directly  as  in  the  general  case.  To  obtain  III,  construct  a 
function  p whose  table  is  given  by  the  following:  when  the  truth-value 
of  p is  that  of  q or  lower,  p d !^q  will  have  the  value  t\,  while  if  the  truth- 
value  of  p is  above  that  of  q,  then  if  the  value  of  p is  t^  or  higher,  p d £ q 
will  have  the  value  of  q,  while  if  it  is  below  t^,  say  tv  and  that  of  q is  tv>, 
then  the  truth- value  of  p d ™ q will  be  /„/_„+ 1.  Ill  will  then  be  simply 

* Or  we  might  take  the  truth-table  as  element  in  which  case  the  system  is  perhaps 
smoother  than  before. 


i\ 


Post:  A General  Theory  of  Elementary  Propositions.  183 

“b  P” 

“\-P 

produce 

“b  Q” 

Now  by  generalizing  each  part  A,  B,  C,  D of  the  proof  of  the  fundamental 
theorem  of  sec.  3 it  can  be  shown  that  by  the  assertion  of  a finite  number 
of  functions  with  values  from  t\  to  p all  such  can  be  obtained.*  If  then  we 
assert  these  functions  in  IV  we  shall  have  every  function  in  the  ^u-space 
asserted.  Furthermore  no  others  can  be  asserted  for  by  the  use  of  II  and 
III  we  can  only  get  functions  with  values  from  t\  to  by  means  of  functions 
similarly  restricted.  This  is  obvious  in  II  while  in  III  if  the  value  of  P is 
from  t\  to  t,,,  while  that  of  Q is  below  t^,  then  from  the  above  definition  of  the 
table  of  P d %i  Q it  would  have  the  value  of  Q and  so  be  below  But  that 
contradicts  the  assumption  that  the  premises  had  values  from  t\  to 

This  set  of  postulates  will  then  give  the  proper  set  of  true  functions. 
Furthermore  let  us  suppose  that  we  assert  a function  with  at  least  one  value 
below  tp.  This  will  contain  a function  <f>(p)  with  but  one  value,  and  that 
below  tp.  By  II,  <f>(p)  will  be  asserted.  Furthermore  since  <h(p)  . 3 £ . <j>{p) 
Dm  has  its  value  t±  it  will  be  asserted,  and  so  we  obtain  by  III 

~ m<t)(p )•  Repetition  of  this  process  will  finally  give  us  a function  \p(p) 
with  but  one  value  tm.  But  \p(p)  • d £ p is  asserted  having  but  one  value  t\. 
We  thus  obtain  the  assertion  of  p.  The  system  is  therefore  closed.  And 
since  all  functions  with  values  from  t^+i  to  tm  are  completely  false,  the  system 
is  completely  closed. 

15.  Comparison  of  Systems. — As  in  the  truth-table  development  we  can 
generalize  the  systems  by  using  arbitrary  functions  as  primitives,  and  as 
was  done  there  we  can  show  how  to  generate  a complete  ?ft-dimensioned 
system  by  one  second  order  function,  and  how  to  give  a completely  closed 
set  of  postulates  for  all  complete  systems.  The  problem  of  determining  all 
possible  systems  of  m-dimensional  truth-tables,  however,  is  one  we  have  not 
considered,  though  its  solution  would  through  considerable  light  on  the 
ordinary  problem. 

We  turn  now  to  the  following 

Definitions.  A closed  system  S with  primitives  fi,  /2,  • • • fn  has  a 
representation  in  a closed  system  S'  with  primitives  f[,  f2,  •••/!/  if  we  can  so 
replace  the  fs  by  functions  in  S'  that  a function  in  S will  be  true  when  and  only 
when  the  correspondent  in  S'  is  true. 

Two  systems  are  equivalent  if  each  has  a representation  in  the  other. 

Denote  a complete  m-dimensional  truth-system  with  the  asserted  func- 
tions forming  a truth-space  of  p dimensions  by  ^ Tm . We  then  have  the 


* Lack  of  space  prevents  us  from  giving  the  details. 


184 


Post:  A General  Theory  of  Elementary  Propositions. 


Theorem.  Two  complete  truth-systems  M Tm  and  M, T 'm,  are  equivalent 
when  and  only  when  p = p'  and  m = m' . 

The  conditions  are  clearly  sufficient  since  we  can  make  truth-values 
correspond.  To  prove  them  necessary  suppose  m > m'.  If  we  construct 
mm  functions  of  first  order  in  T with  different  truth-tables  then  there  will 
be  two,  0i(p),  02 (p)  whose  correspondents  4>[(p),  02 (p)  have  the  same 
truth-tables  since  there  are  in  T'  only  i*C'm  of  first  order.  Let  x(v,  Q)  have 
value  ti  when  p and  q have  the  same  value  and  tm  otherwise.  Then  x(4> b 0i) 
is  true;  hence  x'(4>u  00  is*  02  having  the  same  table  as  0 [,  x'(0i,  00  is 
true,  and  hence  x(0i>  02)  the  correspondent.  But  that  would  make  0i 
have  the  same  table  as  02-  Now  suppose  p > p'.  If  0 have  all  the  values 
from  ti  to  tp  and  no  others  there  are  p * functions  with  values  t\  to  t^  of 
the  form  00  (p).  These  will  then  be  asserted  and  so  the  correspondents 
will  be  asserted  and  have  values  t[  to  t'^.  Since  we  can  only  have  p,tL  func- 
tions 0'0'(p)  with  different  tables,  we  can  find  two  of  the  p M correspondents 
with  the  same  table.  The  above  contradiction  then  results  as  before. 

For  representation  we  have  only  found  the 

Theorem.  To  represent  ^Tm  in  ^T'm,  it  is  necessary  to  have  p — p', 
m ~ m';  it  is  sufficient  to  have  p ^ p' , m — p ^ m'  — p' . 

Corollary.  A necessary  and  sufficient  condition  that  ^ Tm  have  a repre- 
sentation in  fiT'mr,  is  that  m — m'. 

It  is  of  interest  to  note  as  a result  that  the  only  complete  truth-systems 
equivalent  to  the  system  of  ‘Principia’  are  iTYs;  and  though  it  can  be 
represented  in  every  complete  truth-system,  only  iTYs  can  be  represented 
in  it.  We  have  thus  verified  our  statement  that  we  obtain  essentially  new 
logical  systems. 

16.  Interpretation  of  m- valued  Truth-systems  in  Terms  of  Ordinary 
Logic. — Let  the  elementary  proposition  of  the  (~TO,  V«)  system  be  inter- 
preted as  an  ordered  set  of  (m  — 1)  elementary  propositions  of  ordinary 
logic  P = (pi,  P2,  • • • Pm-T)  such  that  if  one  proposition  is  true  all  those 
that  follow  are  true.  P will  be  then  be  said  to  have  the  truth- value  t\  if 
all  the  p’s  are  true,  t2  if  all  but  one  are  true,  etc.  Also  P will  be  said  to  be 
true  if  at  most  ( p — l)p’s  are  false. 

If  P = (pi,  p.2)  • • • Pm- 1),  Q = (qi,  q2,  • • * qm- i)  we  define 

PVmQ.  = . (piVgi,  p2Vg2,  • • • PmV qm)  Df 

^mP  •=  .(^(piVp2V  Pm- 1),  ~(piV  Pm- 1)  • V .£>1  . p2, 

'V  (pi  V * * • Pm- 1)  • V . Pm- 2 • Pm- 1)  Df 

We  easily  justify  these  definitions  by  showing  first  that  PVmQ  and 
P are  “ elementary  propositions  ” when  P and  Q are,  and  secondly  that  they 


Post:  A General  Theory  of  Elementary  Propositions . 185 

have  the  proper  truth  tables.  Thus  in  PVmQ  the  first  pL  V qt  to  be  true  is 
the  first  for  which  either  p or  q is  true ; also  all  later  terms  will  have  p or  q 
true  and  so  will  be  true.  P V mQ  is  therefore  elementary  and  has  the  required 
table. 

But  in  spite  of  this  representation  iT2  still  appears  to  be  the  fundamental 
system  since  its  truth-values  correspond  entirely  to  the  significance  of  true 
and  completely  false,  whereas  in  ^Tm,  m > 2 either  ju  > 1 or  m - /i  > 1, 
and  this  equivalence  no  longer  holds.  We  must  however  take  into  account 
the  fact  that  our  development  has  been  given  in  the  language  of  \T2  and 
for  that  very  reason  every  other  kind  of  system  appears  distorted.  This 
suggests  that  if  we  translate  the  entire  development  into  the  language  of 
any  one  ^ Tm  by  means  of  its  interpretation,  then  it  would  be  the  formal 
system  most  in  harmony  with  regard  to  the  two  developments. 


VITA. 


Emil  Leon  Post ; was  born  at  Augustowo,  Poland,  Feb.  11,  1897 ; ar- 
rived in  America  May,  1904;  received  a B.S.  from  the  College  of  the  City 
of  New  York  1917 ; graduate  student  in  Mathematics,  Columbia  University, 
1917-19 ; A.M.  Columbia,  1918 ; assistant  tutor  in  Mathematics,  College  of 
the  City  of  New  York,  1917-1918;  Instructor  in  Extension  Teaching  and 
Lecturer  in  Mathematics,  Columbia,  1918-1920 ; member  of  4>BK  and  Sigma 
Xi ; author  of  ‘ ‘ The  Generalized  Gamma  Functions,  ’ ’ Annals  of  Mathe- 
matics, 1919. 

While  a graduate  student  in  Columbia  I took  courses  under  Professors 
Fiske,  Cole,  Keyser,  Kasner  and  Fite.  I wish  to  express  my  indebtedness 
to  them  for  their  help  and  inspiration,  and  in  particular  to  Prof.  Keyser, 
whose  ready  sympathy  and  interest  in  my  work  has  been  an  invaluable  aid 
and  encouragement. 


UNIVERSITY  OF  I LLINOIS^RBAN/^^^^^ 


